Stability of the Positive Mass Theorem for Graphical Hypersurfaces of Euclidean Space

Abstract

Abstract. The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in R n+1 . Specifically, for an asymptotically flat graphical hypersurface M n ⊂ R n+1 of nonnegative scalar curvature (satisfying certain technical conditions), there is a horizontal hyperplane Π ⊂ R n+1 such that the flat distance between M and Π in any ball of radius ρ can be bounded purely in terms of n, ρ, and the mass of M . In particular, this means that if the masses of a sequence of such graphs approach zero, then the sequence weakly converges (in the sense of currents, after a suitable vertical normalization) to a flat plane in R n+1 . This result generalizes some of the earlier findings of the second author and C. Sormani [14] and provides some evidence for a conjecture stated there.

On the stability of the positive mass theorem for asymptotically hyperbolic graphs

“…We endow graph[f ] with the metric induced by H n+1 . Following [10] and [17], we make the following definition.…”

“…Remark 2.2. Note that by condition (i) in the previous definition, we can check that lim r→∞ f (r) = C for some constant C, in contrast to the asymptotically flat case in [17] where an asymptotically flat graph is allowed to approach ±∞ as |x| → ∞. In fact, the usual decay conditions to be imposed in the asymptotically flat case in order to ensure that the ADM mass is finite, namely, |∇f | = O(|x| q 2 ) for some q > n−2 2 , imply lim |x|→∞ f (x) = C for dimensions n ≥ 6 (see [19]).…”

“…In this work, we study the stability of the positive mass theorem for asymptotically hyperbolic graphs, adapting to this setting the ideas of Huang and Lee [17], and making use of results of Dahl, Gicquaud and Sakovich [10]. We consider graphs over the hyperbolic space of dimension n, denoted by H n , inside H n+1 .…”

“…To obtain Theorem 1.1 we adapt the methods developed by Huang and Lee in [17] to the asymptotically hyperbolic setting, for which we use the results obtained by Dahl, Gicquaud and Sakovich in [10]. In [10], the mass of asymptotically hyperbolic graphs (possibly with a minimal boundary) in H n+1 is written as the sum of the integral of R(f ) + n(n − 1), where R(f ) denotes the scalar curvature of the graph of f in H n+1 , over the "exterior region" and a boundary integral depending only on the geometry of the minimal boundary.…”

“…We will use the notation in [10] and [17] as much as possible. In addition, the structure of the article is similar to [17]. Namely, in Section 2 we provide all the necessary definitions and terminology.…”